Magnetism Exercise (part 2)

A blue circle has a resitance of $\lambda=1$ per unit length. $\textcolor{#3D99F6}{x^{2}+y^{2}=1}$ .
If the magnetic field due to small segment $(AC)$ is $\vec{B_1}$ and due to large segment $(2π-AC)$ is $\vec{B_2}$
Type your answer as $\vec{B_1}+\vec{B_2}=?$

Details and Assumptions
1) Neglect the magnetic field through black wire.
2) The black wire has $0$ resistance and the battery $(V=1)$ has internal resistance $r_{internal}=1$ .
3) Consider everything in S.I units.
4) Everything is in $xy$ plane.

The problem is original.